7. Which is greater $21: $5, or 3 ft.: 6 ft.? 8. The antecedent is 15, the ratio ; find the conse quent. 9. The consequent is 6. 12, the ratio 25; find the antecedent. 10. The antecedent is of and the consequent is .75; find the ratio. $ II. PROPORTION 176. Proportion (Latin pro, before, + portio, share) is an expression of equality of ratios. 177. The sign of proportion is the double colon (::). NOTE. The sign of equality (=) is often used instead of the double colon. 178. The terms in a proportion are the numbers that make up the proportion. 179. The extremes are the first and fourth terms. 180. The means are the second and third terms. 181. A proportional is any term of a proportion. 182. A mean proportional is a number which is used as the consequent in the first couplet, and as the antecedent in the second. Thus, in 4:8:8:16, 8 is a mean proportional between 4 and 16. 183. Fundamental principles of proportion are: 1. The product of the means is equal to the product of the extremes. PROOF: If a:b::c:d, then tions, ad = bc. a b с = d' and clearing of frac 2. A mean proportional is equal to the square root of the product of the two other terms. PROOF: If a: b::b: c, then b2 = ac (prin. 1). 3. The product of the means divided by either extreme gives the other extreme. 4. The product of the extremes divided by either mean gives the other mean. PROOF: If a:b::c:d, then bc bc, a= d' ad = bc, .. b SIMPLE PROPORTION 184. Simple proportion is an expression of equality between two simple ratios. NOTE. Formerly, proportion was called the Rule of Three, from the fact that three numbers were given to find a fourth. Simple proportion was called Single Rule of Three, and compound proportion, Double Rule of Three. 185. Statement.-Every problem in proportion consists of two parts, a known part and an unknown part. 1. Determine the known part. 2. Determine the 3d term, or base term. 3. Reason from the known to the unknown. Example. If four desks cost $20, what will 7 desks cost? 1. Known part: 4 desks, $20. 66 2. Unknown part: 7 $x. OPERATION: (1) 4 desks: 7 desks :: $20: $x. 7 × $20 ad $35. (Prin. 3, 183.) EXPLANATION.-Write $20 for the 3d term, since it is the same kind as is required in the result. If 4 desks cost $20, 7 desks will cost more than $20; therefore, write the greater number for the 2d term and the lesser for the 1st. The product of the means divided by one extreme gives the other extreme. We cannot multiply $20 by 7 desks; therefore, we use the ratio of 4 desks to 7 desks, which is or 4:7. The terms of either couplet may thus be made abstract. SOLUTION BY ANALYSIS: 66 66 66 1. Since the cost of 4 desks = $20, 66 66 Exercise XIX Find the missing terms in the following: 1. 3:9::7:( ). 2. 45:9:25:( ). 3. 21:3 : :( ):5. 4. 64(): :( ) :1. 9. If 82 bushels of potatoes are raised on acres, how many bushels can be raised on 31 acres? 10. The Washington monument casts a shadow 223 ft. 6.5 in., when a post 3 ft. high casts a shadow 14.5 in. Find the height of the monument. 5. ( ):10:8:16. 6. 7. : : : ( ) : %. 8. 2.5.5 ::.25 :( ). ( ):84: 124. 11. If the interest received on a certain sum of money for 1.5 yr. is $27, how much is the interest on the same sum at the same rate for 2 mo.? 12. A lawyer who collects for 5% gets $34.60 for collecting a debt. Find the amount of the debt. 13. A's property is assessed at $3800. What is his tax at 964 on the $100? 14. A man can do a certain piece of work in 18 days, working 8 hours a day. In how many days can he do the same work by working 10 hours a day? 15. If 36 yards of carpet office floor, how many yards quired to cover it? 16. A man can dig a ditch in 6 days; he and his son can dig it in 4 days. In how many days can the son dig it? 17. If of the value of a ship is $11000, what is of its value? 18. The ratio of A's pay to B's pay is 3. B's pay is $27 per week. What is A's pay per week? COMPOUND PROPORTION 186. A compound proportion is a proportion which contains a compound ratio. The method of reasoning and the principles given in simple proportion apply in compound proportion. of a yard wide will cover my of a yard wide will be re EXAMPLES 1. If 11 men can cut 147 cords of wood in 7 days, working 14 hours a day, how many days will be required for 5 men to cut 150 cords, working 10 hours a day? Or, 1. Known: 11 men, 147 cords, 7 days, 14 hours. 66 66 x 10 OPERATION: 5: 11 147: 150 10: 14 } = :: 7 days: x days. 5 × 147 × 10: 11 x 150 x 14 : 7 days: x days. .. x days = 22 days. EXPLANATION.-7 days is written for the 3d term, because it is the same kind as is required in the result. 1. If 11 men require 7 days, 5 men will require more than 7 days; therefore, the greater number (11) is written for the 2d term, and the lesser (5) for the 1st. 2. If 147 cords require 7 days, 150 cords will require more than 7 days; therefore, the greater number (150) is written for the 2d term, and the lesser (147) for the 1st. 3. If 14 hours a day require 7 days, 10 hours a day will require more than 7 days; therefore, the greater number (14) is written for the 2d term, and the lesser (10) for the 1st. Multiplying the 3d term by the continued product of the 2d, and dividing by the continued product of the 1st, gives 22 days. NOTE.-Employ cancellation wherever possible. SOLUTION BY ANALYSIS: 1. Since time for 11 men working 14 hr. a da. to cut 147 c. = 10 a da. to cut 150 c. = 22 da. NOTE.-After the pupil becomes familiar with each step, he may pass from step 1 to step 4, and then to step 7. By waiting until the last step, numerous cancellations simplify the work. |